Pania, Henare and Matiu are triplets.

Henare, can paint a room by himself in 3 hours.

His sister, Pania can do it in 4 hours.

Their brother Matiu would take 6 hours on his own.

If they all work together and don’t get in each other’s way, how long will the job take?

To solve this problem students need to decide how to set up the equations, what the variables are and what is the relationship between them. It is important that the variables are seen as other than whole numbers.

### Problem

Pania, Henare and Matiu are triplets. Henare, can paint a room by himself in 3 hours. His sister, Pania can do it in 4 hours. Their brother Matiu would take 6 hours on his own.

If they all work together and don’t get in each other’s way, how long will the job take?

### Teaching sequence

- Pose the problem to the class and ask the students to identify the approaches they think might be useful in solving the problem.
- As the students work (in pairs or small groups) check that they are expressing the statements algebraically.
*What variables are in this problem?*

What do we know about the triplets and their painting?

How could you express these? - If the some of the students are having difficulty get others to share the way that they started the problem.
- Share solutions requiring the students to explain their reasoning to the others.
- Look at and discuss the different ways that the students have expressed the equations and their methods of solution.

#### Extension to the problem

Suppose Henare can do another job in 3 hours and Pania can do the same job in 4 hours. Working together, the triplets can now do the whole task in an hour because Matiu has had some help on his technique.

How long would it take Matiu to do the job on his own?

#### Solution

Suppose that Henare paints a fraction h of the room, that Pania paints a fraction p of the room and Matiu paints a fraction m of the room. Then

*h + p + m = 1.*

But this is one equation with three unknowns. What else do we know that can reduce the number of variables for us? We know the relative speeds at which the triplets work. So maybe we can get a relation between h and p.

Look at Henare and Matiu first because the numbers are simpler there. For every 3 hours that Henare paints, Matiu paints 6 hours. So while Henare is doing one room, Matiu is only doing half of a room. If Henare paints half a room then Matiu paints a quarter of the room. So Henare’s fraction is always half as big as Matiu’s. This means that h/2 = m.

If we compare Henare and Pania we can see that, no matter what job of painting they are doing, for every room that Henare paints, Pania will paint ¾ of a room. (This is because in 3 hours, Henare will paint a whole room.) So if Henare paints h of a room, then Pania paints 3h/4 of the room. So p = 3h/4.

So *h* + 3h/4 + h/2 = 1 or 9/4h = 1 or h = 4/9.

But how does this help? Well, Henare takes 3 hours to do the room on his own. This means that he takes 4/9 of 3 hours when he working with his siblings. So he (and the whole job) takes 4/3 of an hour or 1 hr and 20 minutes.

#### Solution to the extension:

In 1 hour Henare can do 1/3 of the job and Pania can do 1/4 of the job. So Matiu does the rest. His fraction is therefore 1 - 1/3 - 1/4 = 5/12 of the job. So if Matiu does 5/12 of the job in an hour he can do the whole job in 12/5 hours. That is, he can do it in 2 hours and 24 minutes.